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Non-Geometric Calabi-Yau Backgrounds and K3 Automorphisms C Non-geometric Calabi-Yau backgrounds and K3 automorphisms C. Hull, D. Israël, A. Sarti To cite this version: C. Hull, D. Israël, A. Sarti. Non-geometric Calabi-Yau backgrounds and K3 automorphisms. Journal of High Energy Physics, Springer, 2017, pp.084. 10.1007/JHEP11(2017)084. hal-01651614 HAL Id: hal-01651614 https://hal.sorbonne-universite.fr/hal-01651614 Submitted on 29 Nov 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Published for SISSA by Springer Received: October 16, 2017 Accepted: November 5, 2017 Published: November 15, 2017 Non-geometric Calabi-Yau backgrounds and K3 automorphisms JHEP11(2017)084 C.M. Hull,a D. Isra¨elb,c and A. Sartid aThe Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, U.K. bLPTHE, UMR 7589, Sorbonne Universit´es, UPMC Univ. Paris 06, 4 place Jussieu, Paris, France cCNRS, UMR 7589, LPTHE, F-75005, Paris, France dLaboratoire de Math´ematiques et Applications, UMR CNRS 7348, Universit´ede Poitiers, T´el´eport 2, Boulevard Marie et Pierre Curie, 86962 FUTUROSCOPE CHASSENEUIL, France E-mail: [email protected], [email protected], [email protected] Abstract: We consider compactifications of type IIA superstring theory on mirror-folds obtained as K3 fibrations over two-tori with non-geometric monodromies involving mirror symmetries. At special points in the moduli space these are asymmetric Gepner models. The compactifications are constructed from non-geometric automorphisms that arise from the diagonal action of an automorphism of the K3 surface and of an automorphism of the mirror surface. We identify the corresponding gaugings of = 4 supergravity in four N dimensions, and show that the minima of the potential describe the same four-dimensional low-energy physics as the worldsheet formulation in terms of asymmetric Gepner models. In this way, we obtain a class of Minkowski vacua of type II string theory which preserve = 2 supersymmetry. The massless sector consists of = 2 supergravity coupled to N N 3 vector multiplets, giving the STU model. In some cases there are additional massless hypermultiplets. Keywords: Differential and Algebraic Geometry, Supergravity Models, Superstring Vacua, Superstrings and Heterotic Strings ArXiv ePrint: 1710.00853 Open Access, c The Authors. 3 https://doi.org/10.1007/JHEP11(2017)084 Article funded by SCOAP . Contents 1 Introduction 1 2 Mathematical background 6 2.1 Lattice-polarized mirror symmetry 7 2.2 Berglund-H¨ubsch mirror symmetry 8 2.3 Non-symplectic automorphisms of prime order 10 JHEP11(2017)084 3 Non-geometric automorphisms of K3 sigma-models 15 3.1 Mirrored automorphisms and isometries of the Γ4,20 lattice 15 3.2 Symmetries of Landau-Ginzburg mirror pairs 18 3.3 Worldsheet construction of non-geometric backgrounds: summary 21 4 N = 4 gauged supergravity from duality twists 22 4.1 Twisted reduction on T 2 23 4.2 Aspects of the low-energy = 2 theory 29 N 4.2.1 Massless multiplets 29 4.2.2 Massive multiplets 30 4.2.3 Accidental massless multiplets 30 4.2.4 Further accidental massless multiplets from KK modes 31 5 Compactifications with non-geometric monodromies 32 5.1 Gravitini masses and supersymmetry 34 5.2 Moduli space 35 6 Conclusion 37 A Explicit lattice computations 38 1 Introduction Geometric compactifications constitute only a subset of string backgrounds and have in- teresting generalisations to non-geometric backgrounds. Examples arise from spaces with local fibrations that have transition functions that include stringy duality symmetries. Spaces with torus fibrations and T-duality or U-duality transition functions are T-folds or U-folds [1], while those with Calabi-Yau fibrations and mirror symmetry transition func- tions are mirror-folds [1]. Such non-geometric spaces often have fewer moduli than their geometric counterparts, and the non-geometry typically breaks some of the symmetries, including supersymmetries, and provide an interesting tool for probing quantum geome- try. Solvable worldsheet conformal field theories (CFT’s) such as asymmetric orbifolds can – 1 – arise at special points in the moduli space of a non-geometric background [2], allowing a complete analysis and important checks on general arguments. Our focus here will be on mirror-folds of the type IIA superstring constructed from K3 bundles over T 2 with transition functions involving the mirror involution of the K3 surface. Previously Kawai et Sugawara have considered in [3] K3 mirror-folds with monodromies that, at least when the fiber is compact, break all supersymmetry; in the present work, we consider in contrast monodromies that preserve 8 supersymmetries, i.e. which preserve a quarter of the 32 supersymmetries of the type IIA string, or a half of the 16 supersymmetries of type IIA compactified on K3. As a result, we find interesting mirror-folds which give D = 4, = 2 Minkowski vacua of type IIA superstring theory. As we shall see, particular JHEP11(2017)084 N examples give precisely the STU model of [4] at low energies. Our constructions can be viewed as particular cases of reductions with a duality twist [2]. In such a construction, a theory in D dimensions with discrete duality symmetry G(Z) (e.g. T-duality or U-duality) is compactified on a d-torus with a G(Z) monodromy around each of the d circles, giving a string-theory generalization of Scherk-Schwarz re- duction [5]. In many cases, the theory in D dimensions has an action of the continuous group G which is a symmetry of the low energy physics, but which is broken to a discrete subgroup in the full string theory. For a field φ transforming under G as φ gφ the ansatz 7→ is of the form φ(xµ, yi) = g(y)φˆ(x) (1.1) where yi, i = 1, . , d, are coordinates on T d and xµ, µ = 0,...,D d 1, are the remaining − − coordinates. With periodicities yi yi + 2πR , the mondromies g(y )−1g(y + 2πR ) must ∼ i i i i be in G(Z) for each i. Of particular interest are the special cases in which there are points in the moduli space in D dimensions that happen to be fixed under the action of the monodromies. For example, consider a theory where the moduli space in D dimensions contains the moduli space for a 2-torus, i.e. SL(2, R)/U(1), identified under the action of the discrete group SL(2, Z). There are special points in the moduli space which are invariant under finite subgroups of SL(2, Z) isomorphic to Zr for r = 2, 3, 4, 6. If one considers reductions on a circle where the monodromy is in one of these Zr subgroups, then at any point in the moduli space invariant under the action of the monodromy, the reduction with a duality twist can be viewed as a Zr orbifold by a Zr twist together with a shift around the circle by 2πR/r [2]. From the effective field theory point of view, the moduli give scalar fields in D 1 dimensions and − the reduction gives a potential for these fields. At each fixed point in moduli space (i.e. at each point that is preserved by the monodromy) the potential has a minimum at which it vanishes, giving a Minkowski compactification [2]. In these cases, at the special points in moduli space there is both a stringy orbifold construction and a supergravity construction, giving complementary pictures of the same reduction. This extends to a reduction on T d to D d dimensions with a duality twist on each of the d circles. The question of finding − reductions giving Minkowski space in D d dimensions becomes related to that of finding − fixed points in moduli space preserved by a subgroup of G(Z). – 2 – Our starting point will be the theory in six dimensions obtained from compactifying IIA string theory on a K3 surface. The moduli space of metrics and B-fields on K3 gives the moduli space of K3 CFT’s and is given by [6, 7] O(4, 20) , O(4) O(20) × identified under the discrete subgroup O(Γ ) O(4, 20) preserving the lattice Γ , 4,20 ⊂ 4,20 which is the lattice of total cohomology of the K3 surface as well as the lattice of D- brane charges in type IIA. O(Γ4,20) is the perturbative duality group of two-dimensional conformal field theories with K3 target spaces, which includes mirror symmetries. We then JHEP11(2017)084 2 reduce to four dimensions on T with an O(Γ4,20) twist around each circle. Generically, such reductions break all supersymmetry; we will focus here on a class of monodromies admitting = 2 vacua in four dimensions. From the string theory point of view, we N will find them by considering points in the moduli space of sigma-models with K3 target spaces that are preserved by finite subgroups of O(Γ4,20), and taking orbifolds by such automorphisms combined with shifts on the circles. The particular automorphisms of K3 CFTs that we will use are inspired by a world- sheet construction of asymmetric Gepner models presented in [8, 9], following earlier works [10, 11] (see [12, 13] for later generalizations).
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